The sum of two numbers is $105$, and their difference is $23$. What are the two numbers?
Explanation: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 105}$ ${x-y = 23}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 128 $ $ x = \dfrac{128}{2} $ ${x = 64}$ Now that you know ${x = 64}$ , plug it back into $ {x+y = 105}$ to find $y$ ${(64)}{ + y = 105}$ ${y = 41}$ You can also plug ${x = 64}$ into $ {x-y = 23}$ and get the same answer for $y$ ${(64)}{ - y = 23}$ ${y = 41}$ Therefore, the larger number is $64$, and the smaller number is $41$.